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The first two entries of a sequence are 3 and 27. Starting from the third entry, every entry is the absolute value of the difference of the two previous entries. What is the 2016th entry of this sequence?

 

After listing it out, I got:

3, 27, 24, 3, 21, 18, 3, 15, 12, 3, 9, 6, 3, 3, 0, 3, 3, 0, 3, 3, 0... and so on, but I'm not quite sure what to do with that.

 

Help appreciated! :D 

 Jan 17, 2021
 #1
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You're on the right track....we have a pattern that you found as

 

Term    1   2   3    4   5   6   7    8    9  10   11  12   13    14  15  16

           3, 27, 24, 3, 21, 18, 3, 15, 12, 3,    9,   6,     3,   [3,    0,   3] , [ 3, 0, 3], ]3, 0 , 3]

 

Notice that......starting with the  14th term   we  have  the repeating pattern   3,0,3

 

Notice that   after  this    ( term   number + 2)   ] / 3       gives us  what we need  to determine the 2016th term

 

Notice  that

 

(14 + 2) / 3  =  16/3 =    5  +   1/3    =    1st   number  in the pattern  =  3

(15 + 2) / 3  =   17/3  =  5   +  2/3  =       2nd number in the pattern  =  0

(16 + 2) / 3  = 18/3  =    6        whole number   result gives us the 3rd number in the pattern   = 3

 

So....to determine the  2016th   term we have

 

(2016 + 2 ]   / 3   =  2018 / 3  =    =   672  + 2/3   =  2nd number in pattern  =   0

 

 

cool cool cool

 Jan 17, 2021

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